Buy firm zucchini with tiny prickly hairs around the stem end; the hairs are a sign of freshness.
1. Adjust one oven rack to upper-middle position and second oven rack to lowest position, then place a rimmed baking sheet on each rack and heat oven to 400 degrees.
2. Meanwhile, halve each zucchini lengthwise. With small spoon, scoop out seeds and most of flesh so that walls of zucchini are 1/4 inch thick. Season cut sides of zucchini with salt and pepper, and brush with 2 tablespoons oil; set zucchini halves cut-side down on hot baking sheet on lower rack. Toss potatoes with 1 tablespoon olive oil, salt, and pepper in small bowl and spread in single layer on hot baking sheet on upper rack. Roast zucchini until slightly softened and skins are wrinkled, about 10 minutes; roast potatoes until tender and lightly browned, 10 to 12 minutes. Using tongs, flip zucchini halves over on baking sheet and set aside.
3. Meanwhile, heat remaining tablespoon oil in 12-inch skillet over medium heat until shimmering but not smoking, about 2 minutes. Add onion and cook, stirring occasionally, until softened and beginning to brown, about 10 minutes. Increase heat to medium-high; stir in garlic and cook until fragrant, about 30 seconds. Add tomatoes and cooked potatoes; cook, stirring occasionally, until heated through, about 3 minutes. Off heat, stir in basil, 1/2 cup cheese, and salt and pepper to taste.
4. Divide filling evenly among squash halves on baking sheet, spooning about 1/2 cup into each, and pack lightly; sprinkle with remaining cheese. Return baking sheet to oven, this time to upper rack, and bake zucchini until heated through and cheese is spotty brown, about 6 minutes. Serve immediately.